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Physical Review A

Integration of the Heisenberg equation of motion for quantum tunneling

Martin Kamela, Mohsen Razavy

Published Mar 1, 1992• 8 citations
Publisher

Abstract

A method of integration of the Heisenberg equation of motion is proposed in which the time evolution of the basis set {${\mathit{T}}_{\mathit{m},}$n(t)} of the Weyl-ordered operators introduced by Bender and Dunne [Phys. Rev. D 40, 3504 (1989)] is obtained from the Taylor expansion and is expressible in terms of the initial {${\mathit{T}}_{\mathit{m},}$n(0)}'s. In the absence of damping forces, the constant values of the energy and the position-momentum commutation relation are used to check the accuracy of the integration. This method is applied to obtain the mean position and velocity of the particle as a function of time as well as the dwell time of the particle inside the barrier. In the example that is considered here, the potential is assumed to be the sum of a harmonic and a cubic term, and the calculation is done with and without dissipative coupling.

DOI
10.1103/physreva.45.2695

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A method of integration of the Heisenberg equation of motion is proposed in which the time evolution of the basis set {Tm,{\mathit{T}}_{\mathit{m},}Tm,​n(t)} of the Weyl-ordered operators introduced by Bender and Dunne [Phys. Rev. D 40, 3504 (1989)] is obtained from the Taylor expansion and is expressible in terms of the initial {Tm,{\mathit{T}}_{\mathit{m},}Tm,​n(0)}'s. In the absence of damping forces, the constant values of the energy and the position-momentum commutation relation are used to check the accuracy of the integration. This method is applied to obtain the mean position and velocity of the particle as a function of time as well as the dwell time of the particle inside the barrier. In the example that is considered here, the potential is assumed to be the sum of a harmonic and a cubic term, and the calculation is done with and without dissipative coupling.
The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set {Sm,n{\mathit{S}}_{\mathit{m},\mathit{n}}Sm,n​(t)} formed from r(t) and pr{\mathit{p}}_{\mathit{r}}pr​(t), one can express the time evolution of any member of the set as an infinite sum involving the operators {Sm,n{\mathit{S}}_{\mathit{m},\mathit{n}}Sm,n​(0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.
A direct integration of the Heisenberg equations of motion yields expansions of the position and momentum operators x(t) and p(t), each as an infinite series in terms of the initial Weyl-ordered basis set {Sm,{\mathit{S}}_{\mathit{m},}Sm,​n} formed from x(0) and p(0), with c-number time-dependent coefficients. This method is applied to the problem of tunneling in a symmetric and an asymmetric quartic potential. The expectation values of the position and momentum operators with minimum uncertainty wave packet 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 can be calculated accurately for a maximum time that is short compared to the period of oscillation of the wave packet. From the result of this calculation, with the help of Prony's method one can determine the level spacings for the low-lying states. In addition, in this formulation the wave packet retains its shape; therefore, one can study the trajectory 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 as the quantal analogue of the motion of the system in phase space.
The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.
The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.
In the second part of this paper on micro-canonical ensemble a new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of the considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the latter function the integral Wigner–Liouville's type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. A numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. Numerical results have been obtained for series of average values of the quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunnelling particles at barrier crossing and the absorption spectra of electrons in a potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunnelling transitions and the effect of the quasi stationary state have been considered as the reasons for the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. The possibility of applying the developed approach to the theory of classical wave propagation in random media has also been considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides.
The quantum tunneling rate of a macroscopic particle in one-dimensional potentials subject to oscillating external fields is considered using the complex time path integral method and the time-dependent Schrödinger equation. A simple calculation scheme for obtaining the tunneling rate within quasi-classical accuracy is described and applied to two systems. The one is the tunneling motion through an inverted parabola potential and the other is the escaping motion from a cubic potential well. Both systems are subject to small oscillating fields. It is found that both of the rates are exponentially enhanced due to the imaginary time motion of the tunneling particle under oscillating potential barriers, and that time scale is identical to the Büttiker-Landauer traversal time.